Rheology of polymer brush under oscillatory shear flow studied by nonequilibrium Monte Carlo simulation

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 123, Rheology of polymer brush under oscillatory shear flow studied by nonequilibrium Monte Carlo simulation Shichen Ji and Jiandong Ding a Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai , China Received 12 May 2005; accepted 13 July 2005; published online 10 October 2005 The rheological behaviors of polymer brush under oscillatory shear flow were investigated by nonequilibrium Monte Carlo simulation. The grafted chain under oscillatory shear flow exhibited a waggling behavior like a flower, and the segments were found to have different oscillatory phases along the chain contour. Stress tensor was further obtained based on the statistics of sampled configuration distribution functions. The simulation reproduced the abrupt increase of the first normal stress difference N 1 with the flow velocity over a critical value, as observed in the experiment of Klein et al. Nature London 352, However, our simulation did not reproduce the brush thickening with shear velocity increased, which was suggested to be responsible for the abrupt increase of N 1 in the above-mentioned paper. This simulation demonstrates that the increase of normal stress might be an inherent behavior of polymer brush due to chain deformation under flow American Institute of Physics. DOI: / I. INTRODUCTION a Author to whom correspondence should be addressed. Electronic mail: jdding1@fudan.edu.cn Grafted polymer layers are useful in many technological applications such as colloidal stabilization and lubrication, and meanwhile constitute an interesting scientific topic. Since the pioneer work of Alexander 1 and de Gennes, 2 numerous papers have been published on the properties of polymer brushes from experimental and theoretical aspects. 3,4 So far the equilibrium properties of polymer brushes have been well known. In contrast to it, the rheological behaviors of polymer brushes under shear, especially under oscillatory shear, are less understood. Using a surface force apparatus, Klein et al. 4 studied elegantly the rheological behaviors of two polymer brushes under oscillatory shear flow. The brushes in a small gap were either compressed or separated. An abrupt increase of the normal force was detected when the relative velocity of the surfaces increased over a critical value, which was interpreted due to brush thickening over a critical flow velocity. 4 On the other side, the direct measurements of brush height under a steady shear have not found such a brush thickening. 5 7 We thus wonder whether the brush thickening can account for the abrupt increase of normal force or not. On theoretical aspects there are different opinions about the brush thickness under shear flow. Rabin and Alexander 8 first examined polymer brush under shear theoretically and found that shear flow did not affect brush thickness. On the contrary, Barrat 9 predicted a brush swelling as large as 25% with respect to the equilibrium brush thickness. His approach is essentially similar to that used by Rabin and Alexander but differs in the different contribution of the osmotic compressibility to the force acting at the free ends. Even larger brush swelling or brush thickening has been reported by Kumaran 10 and Harden and Cates. 11 Today, computer simulation has been regarded as the third research approach besides theory and experiment. Different kinds of molecular simulations have been carried out to study the properties of polymer brush under shear. For steady shear, the significant brush swelling has not been found by most of molecular simulations, such as molecular dynamics MD simulation, 12,13 Brownian dynamics simulation, stochastic simulation, 17 Monte Carlo MC simulation, and dissipative particle dynamics DPD simulation. 21 A few simulations have been carried out to study the behaviors of polymer brush under oscillatory shear, 12,14,15,21 and some of them 14,15,21 focus on the brush swelling of a single brush. Doyle et al. 14,15 studied a polymer brush under oscillatory shear with Brownian dynamics simulation. The abrupt normal force increase was reproduced and accounted for from the increase of osmotic pressure due to shearinduced collisions of beads within the brush. Wijmans and Smit 21 extended DPD simulation to study a brush under oscillatory shear with explicit solvents, but no straightforward relation between brush thickness and associated rheological behaviors was revealed. To our knowledge, no MC simulation of polymer brush under oscillatory shear flow has been reported so far. Our work aimed not just to fill this blank, of course. To date, brush thickening has not yet been supported by direct measurements of brush height under flow, 5 7 and those theoretical analyses and simulation outputs to predict brush thickening 9 11,14,15 were based upon hydrodynamic interaction or osmotic pressure. The present paper would like to employ MC simulation to investigate whether or not a brush will be swollen under flow and whether or not the normal stress will be increased abruptly over a critical shear /2005/ /144904/7/$ , American Institute of Physics

2 S. Ji and J. Ding J. Chem. Phys. 123, rate when complicated hydrodynamic interaction is neglected and osmotic pressure is not explicitly included. MC simulation is a widely used stochastic simulation method. Lattice MC simulation is especially efficient as the computing cost is concerned. The properties of polymer brush under steady shear or without shear have been studied by lattice MC simulation 18,19,22 27 and the results well consist with those from the self-consistent-field SCF theory and other simulations. In MC simulation, the solvent is modeled as a continuum. The hydrodynamic effect cannot be directly involved, unlike some other simulations such as DPD simulation 21 and MD simulation. 28 Fortunately, hydrodynamic effect in a dense system is not so striking as in a dilute solution or a single chain. In our group, the approach of nonequilibrium lattice MC simulation of oscillatory shear has been introduced to investigate the rheology of homopolymer solution 29 and micellar block copolymer solutions. 30 In this paper, lattice MC simulation is extended to study the rheological properties of a polymer brush, which corresponds to the separated brushes under oscillatory shear flow. 4 Besides chain configurations, stress tensor can be obtained from statistics of sampled configuration distribution functions CDFs with the approach put forward in our previous paper dealing with simple shear flow of free homopolymer chains. 31 So, the relation between the abrupt increase of normal stress difference if reproduced and the brush thickening assumed in the literature can be examined. The remainder of the paper is organized as follows. In Sec. II, the lattice model and the simulation approach are described. In Sec. III, the static properties of a polymer brush and the associated nonequilibrium properties under oscillatory shear flow are examined. A brief summary is given in Sec. IV. FIG. 1. Schematic presentation of a polymer brush. All of the grafted chains are assumed with the same chain length N. y denotes the averaged occupying fraction of segments in the plane at a given y. The averaged position of the free ends y N is employed to define brush thickness. The tilting angle describes the orientation of the vector from the grafting point to the center of mass of the associated grafted chain. An oscillatory flow field along the x dimension is exerted upon the polymer brush with a velocity gradient along the y dimension. N is fixed as 40. The grafting density 0, defined as the occupying fraction of the grafting ends in the grafting plane, was set as unless otherwise indicated. The very initial state of polymer brush was created as follows: After randomly generating completely stretched grafted chains, the system was run for a sufficiently long time without shear to reach an equilibrium state which serves as the initial state for further simulation. In this study time t was measured in units of Monte Carlo step MCS, which is defined as the attempt number during which all the beads in the system are tried once on average. II. MODEL AND SIMULATION APPROACH A. Lattice chain model The simulation of self-avoiding chains was carried out in a simple cubic lattice system L x =L z =20, L y =40. The periodic boundary condition is applied in the x and z directions. The x direction is defined as the flow direction and the z direction is the neutral direction, while the y direction is along the velocity gradient. An impenetrable grafting surface is set at y=0 xz plane. The chosen size in the y direction is sufficiently large to avoid the constraint of the grafted chains along this dimension. The Larson-type bond fluctuation model with the permitted bond length as 1 or 2 is employed and the associated microrelaxation modes include chain twisting. 32,33 The coordination number of each bead in three dimensions is 18. Any lattice site cannot be occupied by two or more beads simultaneously, and bond intersection is forbidden in any elementary movement. The excluded volume effect has been taken into account completely further considering the impenetrable wall effect. Due to this effect, grafted chains with a sufficiently high grafting density constitute a polymer brush with a stretched chain configuration. In the present paper, a single brush layer composed of multiple grafted chains with N beads per chain was simulated, as schematically presented in Fig. 1. The chain length B. Introduction of shear flow In this nonequilibrium MC simulation, the shear field is introduced via a biased jump rate, i.e., the jump rate is higher in the flow direction and lower in the opposite direction. For simplicity, the inputted shear rate is assumed to be spatially uniform at any given time. Then, when a bead with coordinate x 1,y 1,z 1 moves to a new site x 2,y 2,z 2, the jump rate P under oscillatory shear field is P = P ȳ x cos t, 1 where P 0 is the jump rate without shear field and equals 1/18 18 comes from the coordination number of each bead, 0 is the inputted amplitude of the oscillatory reduced shear rate, y is the distance from the bead to the grafting wall, ȳ is the average distance of the old and new positions in an elementary movement, x is the displacement in an elementary movement of the selected bead along the flow direction, and is oscillatory shear frequency. In Eq. 1, itisȳ instead of merely y 1 or y 2 that should be used as suggested in our previous paper to study simple shear flow of free homopolymers. 31 Such a treatment is necessary to meet the requirement of the microscopic reversibility as indicated by Milchev et al. in the off-lattice MC simulation of homopolymers subject to simple shear flow. 34 In the simulation, P

3 Rheology of polymer brush J. Chem. Phys. 123, should be larger than zero, which determines the maximum available value of 0 L y. C. Statistics of stress tensor In our model the macroscopic stress tensor can be calculated by statistics of the sampled CDFs. This method has been successfully applied to study the rheological properties of both homopolymer and block copolymer solutions ,35 Following our previous work, 29 31,35 the reduced stress tensor ˆ of the bond-fluctuation model in the Kramers form is employed as ˆ = ˆ ˆ0 = n p V N 1 Î N 1 K r i+1 r i r i+1 r i, i=1 2 where ˆ and ˆ0 are the total stress tensor of polymer solution and that of pure solvents; is the reciprocal of the Boltzmann constant times absolute temperature; Î is the secondrank unit tensor; n p is the number of chains in the lattice system with volume V; N is the number of beads per chain; r i is the position vector of the ith segment; and K is the elastic constant of the fluctuated bond spring, which is set to be 1 and does not influence the mechanical behaviors of polymer brush. The ensemble average is made over chain configurations and thus the sampled CDFs. It is known that the excluded volume effect influences the calculation of stress tensor. 36 In this simulation, the excluded volume interaction affects the CDFs and thus the resultant stresses. Once the stress tensor is known, the shear stress and the first and second normal stress differences N 1 and N 2 can be obtained from = xy, N 1 = xx yy, N 2 = yy zz. 3 Since the chains in a polymer brush deform strikingly even at the static state, a nonzero background must exist in the calculation of normal stress according to Eq. 2. Such a background has been deduced in the presentation of normal stress of grafted chains subject to flow. III. SIMULATION OUTPUTS AND DISCUSSION A. Static properties of polymer brush The static properties of polymer brushes have been extensively studied in the literature. Our results are just briefly presented. Figure 2 shows the segment density y at a distance y from the grafting surface. y is defined as the average occupying fraction of segments in the plane at the given y. y exhibits a maximum near the surface due to the depletion of the hard wall. The brush thickness is defined as the average position of the free ends y N, which scales with 0 with exponent of 1/3 in satisfactory agreement with the scaling analysis of a polymer brush in good solvent 1,2 and the SCF theory. 37,38 The distribution of the segments in the y direction is rather broad Fig. 3. The free ends i/n=1 can, although located outmost on average, distribute throughout the brush. FIG. 2. a Segment density y at a distance y from the grafting surface with different grafting density 0 ; b Brush thickness y N as a function of grafting density 0. The chain length N=40; 50 independent trajectories were averaged. Theoretical scaling with exponent of 1/ 3 is also marked. B. Nonlinear velocity profile of oscillatory shear flow As described in Sec. II, the inputted flow field is introduced by a biased jumping rate. The outputted velocity profile of polymer brush is first examined. The velocity of polymer at each layer is defined as the average displacement of all tried segments at this layer along the flow direction per MC step. As seen in Fig. 4 a, the velocities of given layers sinusoidally change with MC time t MC as expected in an oscillatory shear flow. The velocity profile inside the brush at a given time is shown in Fig. 4 b. Though a linear inputted shear field at any given time has been introduced, the output velocity is not linear any more. The deviation from a linear velocity profile may be related to the inhomogeneous density distribution of the segments at different layers due to the excluded volume effect as indicated in Fig. 2. The high segment density at the inner of the brush hinds the mobility and thus the resultant velocity. FIG. 3. Spatial distribution of the indicated typical monomers. i y is the probability to find segment i at the layer y from the grafting surface. Grafting density 0 =0.125.

4 S. Ji and J. Ding J. Chem. Phys. 123, FIG. 5. The inclination cos of polymer chain as a function of MC time t MC with given amplitudes of reduced shear rate. The other simulation parameters are the same as those in Fig. 4. FIG. 4. a The velocity profile v x y of give layers as a function of MC time t MC. b The velocity profile v x y as a function of position y at a given time. The dashed lines are used just to guide eyes. N=40; 0 =0.125; the amplitude of the reduced shear rate 0 =0.004; the oscillatory period is 2000 MCS; 500 independent trajectories were averaged; before statistics, several oscillatory periods have been run to make the system in a repetitive state under the exerted oscillatory flow field. Meanwhile, the shear field still penetrates a larger portion of the brush compared with the prediction of the SCF theory. 39 Neelov et al. 17 have found that many of the observed features of polymer brushes under steady shear are rather universal and do not depend on the details of the shear field. Thus, the simple approach in the present paper might be suitable to study polymer brush under shear flow. A nonlinear inputted shear field, which can be calculated according to the Brinkman equation 40 based on the static properties of brush, might be used to modify the velocity profile in the future. C. Chain configuration analysis As the abrupt increase of the normal stress was found at high frequencies, 4 the present paper is focused upon the properties of a polymer brush under oscillatory shear with a period less than the longest relaxation time or Rouse time N. N of a free chain can be readily calculated according to the normal coordinate analysis. 41,42 The Rouse time of an isolated free chain with N=40 was determined as MCS in our MC simulation. The real relaxation time for a chain in the polymer brush must be larger than this value. So, the default oscillatory period of 2000 MCS in our simulation guarantees a high frequency. The shear flow field causes the chains to tilt in the shear direction. The inclination of chains is defined as cos, where is the angle between the shear direction and the direction of the vector from the grafting point to the center of the mass of the chain. According to Fig. 5, the grafted chains exhibited sinusoidal inclination with time. The maximum inclination increases reasonably with increasing shear rates. A typical profile of the different components of the squared radius of gyration S 2 is shown in Fig. 6. The y component of the squared radius of gyration S y 2 is much larger than the x component S x 2 and the z component S z 2, since the chains in the brush have been stretched perpendicular to the grafting surface due to the excluded volume effect. S x 2 sinusoidally changes with t MC with a double frequency compared with the shear field. The changes of S y 2 and S z 2 are relatively less striking and with different phases compared with the change of S x 2. The probability of finding a free end at the position x, where x is the displacement relative to corresponding grafting point of a chain, is denoted as e x. The probabilities at four given times are shown in Fig. 7. A broad Gaussian-type distribution of e x was seen. It is interesting that the peak values and also the averages at 0 MCS zero phase angle with the inputted oscillatory shear flow field and 1000 MCS phase angle with respect to the flow from right to left are not zero, but positive and negative, respectively. It seems that the free ends surpass the other segments in response to the exerted flow field, which is further confirmed by Fig. 8. Typical averaged chain configurations, at a time interval of 100 MCS during one oscillatory period of 2000 MCS, are obtained by plotting y i as a function of x i as shown in Fig. 8 a. The simulation reveals that the grafted chains subject to oscillatory shear flow behave like waggling flowers. Furthermore, different segments along the chain contour do not re- FIG. 6. Brush thickness y N and different components of the squared radius of gyration S 2 as a function of t MC. The amplitude of the reduced shear rate 0 =

5 Rheology of polymer brush J. Chem. Phys. 123, FIG. 7. The distribution probability of the free ends in the shear direction e x at marked simulation times. Here, x is the displacement of the free end relative to corresponding grafting point of each chain along the flow direction. 0 = spond to the exerted flow field with the same phase. The phase difference is especially striking for those segments close to the free ends or in the tail as indicated in the inlet in Fig. 8 a. So, the temporal pattern of grafted chains under oscillatory shear flow looks like a cockscomb flower other than a fan. Even the shear field is synchronously exerted on the segments, the segments in the tail respond faster than those of the inner. The other segments seem to be dragged by the free end. The configurations of grafted chains at different FIG. 9. The first and second normal stress differences N 1 and N 2 and shear stress as a function of t MC. 0 = amplitudes of reduced shear rates are shown in Fig. 8 b. It seems that more segments have been dragged by the free end at higher shear rates. D. Normal stress analysis The macroscopic stress tensor of the polymer brush under oscillatory shear was further calculated by statistics of the sampled CDFs. For homopolymer solutions, the first normal difference N 1 is positive and the second normal difference N 2 is negative as found in both the experiments 43,44 and our previous simulation. 29 For the polymer brush, N 1,N 2, and shear stress as a function of MC time t MC are shown in Fig. 9. Reasonably, N 1 exhibits a double-frequency oscillation while shear stress exhibits the same frequency as that of the exerted flow field, similar to the behaviors of homopolymer solutions. 43,44 The average first normal difference N 1 during oscillatory period as a function of different amplitudes of reduced shear rates is shown in Fig. 10. The abrupt increase of N 1 observed in experiment 4 was reproduced by our simulation. At low shear rates, the fluctuation of N 1 is very large, and the mean value N 1 is almost a constant. When shear rates are high sufficiently, N 1 increases significantly. The increase of averaged brush thickness y N is, however, not reproduced Fig. 11. For our simulation system, the brush thickening assumed by Klein et al. 4 cannot be used to account for the abrupt increase of normal stress. It is known that for a polymeric system the macroscopic stress is inherently related to the chain deformation ,35 In our opinion, it is the large deformation of brush that causes the abrupt increase of N 1. In the case of a rheological sys- FIG. 8. a Flower-waggling-like configuration of grafted chains under oscillatory shear flow. The averaged segment positions with time interval of 100 MCS during one shear period 2000 MCS are shown, which are marked with two kinds of symbols for opposite shear directions. The i denotes segment number. For a clear view, during the flow process from left to right, only odd-numbered segments are shown; from right to left, only even-numbered segments are shown. Different segments exhibit different phases in response to the exerted shear field. The inlet shows clearly a response delay of chain tails after the flow direction was altered from right to left. b The configuration of the chain tails at the time with zero phase of the inputted oscillatory flow field with different amplitudes of reduced shear rates. FIG. 10. The average first normal difference N 1 during oscillatory period as a function of the amplitudes of the reduced shear rate 0.

6 S. Ji and J. Ding J. Chem. Phys. 123, FIG. 11. The averaged brush thickness y N and the root of the different components of the averaged squared radius of gyration S 2 1/2 as a function of the amplitudes of the reduced shear rate 0. FIG. 13. The average first normal difference N 1 as a function of the maximum displacement of free ends x N max with given oscillatory periods. tem composed of polymer chains, the physical origin of the first normal stress difference is the difference of the chain deformation along the flow direction and that along the velocity gradient direction. We further calculated the different components of the averaged squared radius of gyration S 2 during oscillation under different shear rates; the results are also shown in Fig. 11. S x 2 increases with increasing shear rate over a critical value of shear rate, while S y 2 decreases. Hence, the abrupt increase of N 1 is a nonlinear behavior of polymer brush subject to shear flow. If this opinion is valid, the increase of N 1 might also be observed at low frequencies with sufficiently large brush deformation. The maximum deformation of brush in this simulation is accordant with the maximum displacement of the free ends in the x direction x N max during oscillatory shear flow. x N max seems proportional to the product of the amplitude of reduced shear rate 0 and the oscillatory period T, as illustrated in Fig. 12. We further examined the effect of oscillatory frequency on N 1. The first normal stress difference was plotted against x N max at different shear periods Fig. 13. Our simulation reveals that the abrupt increase of N 1 could be observed even at low frequencies, i.e., large periods. However, a large x N max is required to observe the abrupt increase of N 1 at lower frequencies. Hence, a large deformation is required to cause the nonlinear behavior of brush at a lower frequency. Since the experiment of the separated brushes was done only at one frequency, 4 more experiments at different oscillatory frequencies are called for to test our simulation outputs. IV. CONCLUSIONS Nonequilibrium MC simulation has been carried out to study the rheological behaviors of a polymer brush under oscillatory shear flow. The grafted chains in the polymer brush subject to oscillatory shear flow behave like waggling flowers. At high frequencies, segments exhibit different phases and the chain movement seems to be led by the free end. To better understand the rheological properties of polymer brush under oscillatory shear flow, the stress tensor of polymer brush has been calculated based on the CDFs. The increase of the average first normal stress difference with increasing shear rate has been reproduced. However, no brush thickening has been observed in our simulation. So, for our simulation system, it seems not absolutely necessary to introduce brush thickening to interpret the abrupt increase of normal stress with the increase of shear. The nonlinear behavior at large chain deformation might be sufficient to interpret the macroscopic changes of rheological behaviors. According to the present simulation, the increase of N 1 might also be found at lower frequencies but with larger necessary displacements of grafted chains, unless the frequency is too low to induce a nonlinear deformation. Due to the feature of lattice MC simulation, only the properties of a single brush have been examined. The hydrodynamic effect, which may significantly affect the properties of brush under shear, is not included. Since the abrupt increase of normal stress differences has been reproduced in this simplified simulation approach, this rheological phenomenon might not be really unusual in principle, but belong to an inherent behavior of a polymer brush. Our simulation provides a new opinion about the observed increase of N 1, but it still needs to be tested by further experiments with improved sensitivity to normal stress at more frequencies. ACKNOWLEDGMENTS FIG. 12. The maximum displacement x N max of the free end relative to the grafting point of each chain in the x direction as a function of the product of oscillatory period T and amplitude of the inputted reduced shear rate 0. The authors are grateful for financial support from NSF of China Grant Nos , , , , and Two-Base Grant, the Key Grant of Chinese Ministry of Education Grant No , the Award Foundation for Young Teachers from Ministry of Education, 973 project, 863 project Grant No. 2004AA215170, Science and Technology Developing Foundation of Shanghai, and Senior Visiting Scholar Foundation of Key Laboratory.

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